Heat Equation Tricky Question

Hi. Having problems with this tricky Heat Equation Question. Managed to do part (a) and would appreciate verification that it's right.
But I can't manage to finish off the second part. I've started it off so please do advice me. Thanks a lot!
QUESTION:
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(a)

Show that the steady solution (which is independent of t) of the heat equation,

where is a constant, on the interval: with conditions: is:

(b)
Use methods of separation of variables to show that the unsteady solution for with conditions: :

My attempt:

Part (A)::
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Using separation of variables:

now to use the conditions:
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That's part (A) done. is my method to approach the final answer correct?

Show that the steady solution (which is independent of t) of the heat equation,

where is a constant, on the interval: with conditions: is:

We look for a solution of the form .
Where solves the homogenous boundary value problem and is the steady-state function.
We want to get, and .
With with conditions, and .
Since we want and .
Solving this system of equations we get, .

Use methods of separation of variables to show that the unsteady solution for with conditions: :

The only thing you need to know here is that if you are solving an equation with non-homogenous boundary conditions and you seperate it into a sum of two solutions one of which is a steady-state solution then the full solution to the original problem would be the sum of the steady-state function and the solution to the equation with homogenous boundary conditions. In this case the steady-state function (similar to first part) is . Then you need to solve the equation for . Where with and . But this is a homogoneous heat equation. Its solutions is . And the are determined by integral forumals while solving this problem.

But this is a homogoneous heat equation. Its solutions is . And the are determined by integral forumals while solving this problem.

Hi. Thanks.
I understand that once I get then I have to apply the fourier series to get A_n.

I can get close to the summation formula you have just stated, but I seem to have 2 terms inside the summation instead of JUST ONE.

Here is what I got, please shed some light on where to take this.

This is how I've been taught to work out the solution: I'll try to explain in as much detail as possible.

- First I try to simplify the general form of the equation and try and turn it into a "summation" kind of form so that it is ready to have the FOURIER SERIES applied to it. So here is what i have so far:

First I have been taught to always use the general equation form for an unsteady solution which is:
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We then apply the 3 conditions on the general form to try and convert the general solution to something that resembles a summation form, so that we can apply the fourier series to it.

now to use the First condition. Please note that I have absorbed the "C " from the general equation into the other constants:
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- What usually happens with these questions is that the first condition says instead of = T.
- If it was = 0, then the whole solution would have resulted in A=0.
- So going back to the GENERAL SOLUTION, I would have been able to cancel the term with the A becauase "0" makes the whole term go to "0".
- Instead we have A as so I can't cancel the term out and instead have to work with both terms of the general solution when applying the second condition.

The only thing you need to know here is that if you are solving an equation with non-homogenous boundary conditions and you seperate it into a sum of two solutions one of which is a steady-state solution then the full solution to the original problem would be the sum of the steady-state function and the solution to the equation with homogenous boundary conditions. In this case the steady-state function (similar to first part) is . Then you need to solve the equation for . Where with and . But this is a homogoneous heat equation. Its solutions is . And the are determined by integral forumals while solving this problem.